Exact nonrelativistic polarizabilities of the hydrogen atom with
the Lagrange-mesh method
Daniel Baye∗
Physique Quantique, C.P. 165/82, and
Physique Nucléaire Théorique et Physique Mathématique, C.P. 229,
Université Libre de Bruxelles (ULB), B 1050 Brussels, Belgium
(Dated: December 20, 2012)
Abstract
Exact analytical expressions of the dipole polarizabilities of the non-relativistic hydrogen atom
in spherical coordinates are derived with the help of the Lagrange-mesh numerical method. This
method can provide exact energies and wave functions for well chosen conditions of calculation.
Exact dipole polarizabilities are obtained after an unambiguous rounding up to at least principal
quantum numbers around n = 30. The scalar polarizability of any nl level is given by n4 [4n2 +
14 + 7l(l + 1)]/4 and its tensor polarizability by −n4 [3n2 − 9 + 11l(l + 1)]l/4(2l + 3), which allow
calculating the polarizability of any hydrogen state nlm.
PACS numbers: 31.15.ap, 32.10.Dk, 02.70.Hm, 02.70.Jn
∗
[email protected]
1
I.
INTRODUCTION
The non-relativistic hydrogen atom is one of the most studied and best known problems of
quantum mechanics [1]. Since the wave functions are known exactly, many of its properties
can be derived analytically. An interesting exception is given by static dipole polarizabilities
for which a general analytical expression as a function of the quantum numbers nlm related
to spherical coordinates is not available. A polarizability is the response of the electron cloud
to some external field [2, 3] that can be represented as a multipole operator. Polarizabilities
appear into e.g. the Stark effect, interactions between an electron and an atom and van der
Waals forces.
The dipole polarizability of the ground state of the hydrogen atom is known for a long
time [4]. In parabolic coordinates, the general analytical expression for the hydrogen atom
is well known [3] as
1
α(n1 n2 m) = n4 [17n2 − 3(n1 − n2 )2 − 9m2 + 19]
8
(1)
in atomic units, where n1 n2 m are the parabolic quantum numbers and the principal quantum
number is given by n = n1 + n2 + |m| + 1. Thedy allow calculating the Stark Stark effect
at the second oorder of perturbation theory, I.E when the electric field is not too large
but large enough so that the fine-structure splitting can be neglected. This polarizability
involves states of mixed parity resulting from the famous degeneracy of hydrogen. Firstorder perturbation theory mixes the degenerate states which enter into the second-order
polarizabilities.
For spherical quantum numbers nlm, the situation is different. Polarizabilities α(nlm) can
be defined mathematically for each unmixed state nlm. They are useful in different respects.
They provide exact limits of dynamical polarizabilities when the frequency tends to zero [5].
They are a useful testing ground for numerical techniques [6, 7], in particular because of
the similarity of simple models of alkali atoms with hydrogen. However, contrary to the
alkali case, the hydrogen atom polarizabilities are unrealistic in the sense that degenerate
states must be excluded. Rigourous treatments must take into account transitions towards
states with the same quantum number n separated by the small spin-orbit and Lamb-shift
splittings [8]. The non-relativistic polarizabilities are then useful as a good approximation
of the part of the relativistic polarizabilities involving different principal numbers.
2
For ns states, a general analytical result has been derived by McDowell [9],
1
α(ns) = n4 (2n2 + 7).
4
(2)
For |m| = n − 1, l is also equal to n − 1 and one deduces from Eq. (1) [3]
1
α(n,l=n−1,|m|=n−1) = n4 (n + 1)(4n + 5).
4
(3)
General expressions have been derived by Krylovetsky, Manakov, and Marmo but are not
easily accessible (see references in Ref. [5]. For m = 0, they obtain
α(nl0 = n4
n2 [19l(l + 1) − 12] + 13l(l + 1)[3l(l + 1) + 2] − 42
.
4(2l − 1)(2l + 3)
(4)
Otherwise, explicit values of static polarizabilities of the hydrogen atom are given for a
limited number of states in Refs [6, 7].
The Lagrange-mesh method is an approximate variational method involving a basis of
Lagrange functions and using the associated Gauss quadrature for the calculation of matrix
elements [10–12]. Lagrange functions are continuous functions that vanish at all points
of a mesh but one. The principal simplification appearing in the Lagrange-mesh method
is that the potential matrix is diagonal and only involves values of the potential at mesh
points. The kinetic-energy matrix can be calculated exactly or approximated with the Gauss
quadrature. The remarkable property of the Lagrange-mesh method is that the accuracy of
the variational method is essentially preserved in spite of the use of the Gauss quadrature
[13].
Strikingly, for well chosen conditions of the calculation, the Lagrange-mesh method with
exact expressions for the overlaps and kinetic-energy matrix elements can provide the exact
values of the hydrogen energies and wave functions, and thus, as explained below, of polarizabilities. Since computers have a limited number of digits, the numerical results differ
from the exact values by rounding errors.
The aim of this paper is to derive exact analytical dipole polarizabilities by fitting numerical results obtained with a variational calculation using a Lagrange basis, which is exact in
principle. The rounding required because of the limited accuracy of the computer can be
performed unambiguously up to high values of n. In passing, it is observed that a calculation
with the simplest version of the Lagrange-mesh method leads to more accurate numerical
values of the polarizabilities than the variational calculation, in spite of additional Gauss
quadrature approximations.
3
In Sec. II, general expressions for the polarizabilities induced by multipole fields are
summarized and analyzed. In Sec. III, the Lagrange-Laguerre basis is introduced and the
corresponding variational calculation is shown to lead to exact results. The difference with
the Lagrange-mesh method is explained. Numerical results for the dipole polarizabilities are
presented in Sec. IV and their exact analytical expressions are deduced. Concluding remarks
are presented in Sec. V.
II.
POLARIZABILITIES OF THE HYDROGEN ATOM
The wave functions of the non-relativistic hydrogen atom are separated in spherical coordinates as r−1 ψnl (r)Ylm (Ω). The radial functions ψnl with principal quantum number n
are eigenfunctions of the radial Hamiltonian of the hydrogen atom for partial wave l
1
d2
1
l(l + 1)
Hl =
− 2+
−
2
2
dr
r
r
(5)
(in atomic units) and correspond to energy −1/2n2 .
The polarizabilities are often defined by series involving the continuum. These series can
be summed in a compact form with the method of Dalgarno and Lewis [14]. Let us consider
p
(λ)
(λ)
(λ)
the polarization by a multipole operator rλ Cµ where Cµ (Ω) = 4π/(2λ + 1)Yµ (Ω). For
(1)
partial wave l, the radial functions ψnll′ at the first order of perturbation theory are solutions
of the inhomogeneous radial equations
(1)
(Hl′ − E) ψnll′ (r) = rλ ψnl (r)
(6)
where ψnl is the radial wave function of the studied state. The polarizability of state lm for
component µ of the multipole operator is given by
(nlm)
αλµ
= (2l + 1)
X
l′


l′
l λ
m µ −m − µ
2
′
 αλ(nll ) .
(7)
This expression allows a calculation for any µ and m from 3jm coefficients and λ+1 reduced
polarizabilities. These reduced polarizabilities read
(nll′ )
αλ
′

= 2(2l + 1) 
l′ λ l
0 0 0
4
2

Z
∞
0
(1)
ψnll′ (r)rλ ψnl (r)dr.
(8)
The sum in (7) is thus restricted to |l − λ| ≤ l′ ≤ l + λ with l + λ + l′ even and l′ ≥ |m + µ|.
The average or scalar polarizabilities defined by
(nl)
αλ
l
X
1
(nlm)
=
αλµ
2l + 1 m=−l
(9)
do not depend on µ and can easily be calculated with
(nl)
αλ
l+λ
X
′
1
′ (nll )
αλ
=
2λ + 1 ′
(10)
l =|l−λ|
where the prime means that the sum runs by steps of two and contains in general λ + 1
terms.
The right-hand side of Eq. (6) is a polynomial of degree n + λ multiplied by exp(−r/n).
It behaves near the origin as rλ+l+1 . The solution of such an equation is elementary. It
is a polynomial of degree n + λ + 1 multiplied by the same exponential. It behaves near
′
the origin as rl +1 . Hence the integrand in Eq. (8) is a polynomial of degree 2n + 2λ + 1
multiplied by exp(−2r/n) and the integral can be calculated exactly.
An unusual problem arises in the calculation of the static polarizabilities for the hydrogen
atom because of the degeneracies in the level scheme. The differential equation (6) has no
solution when transitions are possible toward a degenerate state. For dipole polarizabilities,
this occurs when l′ = l − 1 and |m + µ| < l′ or when l′ = l + 1 and n > l + 1. Degenerate
energies must be excluded from the calculation of polarizabilities. By projecting out the
degenerate state [7], Eq. (6) is modified into
(1)
(Hl′ − E) ψnll′ (r) = rλ ψnl (r) − hψnl′ |rλ |ψnl iψnl′ (r).
(11)
(1)
The reduced polarizability is obtained with Eq. (8) where ψnll′ is the solution of Eq. (11),
orthogonalized to ψnl′ . For particular cases, analytical calculations are quite feasible but the
structure of a general formula is not obvious.
III.
LAGRANGE-MESH AND VARIATIONAL METHODS
The Lagrange-Laguerre mesh points xi are defined for i = 1 to N by [10]
LN (xi ) = 0,
5
(12)
where LN (x) is a Laguerre polynomial of degree N . The corresponding Gauss-Laguerre
quadrature
Z
∞
0
g(x) dx ≈
N
X
λk g(xk )
(13)
k=1
involves the Gauss-Laguerre weights λk [15]. It is exact if g(x) is a polynomial of degree at
most 2N − 1 times exp(−x) [16]. The regularized Lagrange functions read [12, 17]
−1/2
fi (x) = (−1)i xi
xLN (x) −x/2
e
.
x − xi
(14)
These basis functions are polynomials of degree N multiplied by exp(−x/2). They vanish
at the origin. They satisfy the Lagrange property
−1/2
fi (xj ) = λi
δij .
(15)
At the Gauss approximation, the matrix elements of a function V (x) are diagonal,
Z ∞
N
X
fi (x)V (x)fj (x) dx ≈
λk fi (xk )V (xk )fj (xk )
0
k=1
= V (xi )δij
(16)
because of the Lagrange conditions (15). Matrix elements of x−1 and x−2 are exact since
they involve polynomials of respective degrees 2N − 2 and 2N − 1, but not those of 1 since
they involve a polynomial of degree 2N . This basis is thus not orthonormal [13],
(−1)i−j
Nij = hfi |fj i = δij + √
xi xj
(17)
where the Dirac notation represents an integral from 0 to ∞. The matrix elements Tij =
hfi |T |fj i of T = −d2 /dx2 read at the Gauss approximation [17]
Ti6G=j = (−1)i−j √
xi + xj
xi xj (xi − xj )2
(18)
and
TiiG = −
1 2
xi − 2(2N + 1)xi − 4 .
2
12xi
(19)
The superscript G indicates the use of the Gauss approximation in a case where it is not
exact. Here it is not exact since the integrand is a polynomial of degree 2N times exp(−x).
However, the exact matrix elements of T can be obtained as
1
Tij = TijG − (−1)i−j √
4 xi xj
6
(20)
with a simple correction to the Gauss approximation [11, 13].
The regularized Lagrange-Laguerre basis plays a special role for the hydrogen atom since,
for some well chosen scaling parameter h, the exact radial wave functions can be expressed
exactly as a finite combinations of these Lagrange functions. The radial function ψl are
expanded as
ψl (r) = h
−1/2
N
X
cli fi (r/h).
(21)
i=1
This expression is a polynomial of degree N times exp(−r/2h). For h = n/2, the functions
(1)
ψnll′ (r) and ψnl (r) can be reproduced exactly by expansion (21) if N is large enough.
The exact variational equations for orbital momentum l with basis (21) read
N
X
(Hlij − ENij )clj = 0
(22)
1
l(l + 1)
1
−
δij
= 2 Tij +
2h
2h2 x2i
hxi
(23)
j=1
for i = 1 to N , where
Hlij
is the matrix element hfi |Hl |fj i calculated exactly with Eq. (20) for the kinetic energy.
A crucial property is that the Gauss quadrature is exact for the Coulomb and centrifugal
terms. The system (22) is then an exact variational calculation with a basis chosen in a
space supporting the exact solution for n = 2h if 2h is an integer. Each eigenvalue −1/2n2
can be reproduced exactly but for different values of the scaling parameter h. When the
energy is exact, the corresponding expansion (21) is exact. In this case, the notations En ,
cnl and ψnl referring to the exact wave function can be used.
(1)
Like in Eq. (21), the functions ψnll′ are expanded as
(1)
ψnll′ (r)
=h
−1/2
N
X
(1)
cnll′ j fj (r/h).
(24)
j=1
Eq. (6) leads to the algebraic system
N
X
j=1
(1)
(Hl′ ij − En Nij )cnll′ j = (hxi )λ cnli ,
(25)
where Hl′ ij is the matrix element (23) with l replaced by l′ . For h = n/2, it can give the
exact solution corresponding to ψnl if N is large enough. With the Gauss quadrature, Eq. (8)
7
provides the reduced polarizabilities
(nll′ )
αλ

′
= 2(2l + 1) 
l′ λ l
0 0 0
2
 hλ
N
X
(1)
cnll′ j xλj cnlj .
(26)
j=1
(1)
The Gauss quadrature is exact for N ≥ n+λ+1. Since in this case the cnlj and cnll′ j provide
(1)
the exact functions ψnl and ψnll′ , the polarizabilities are exact.
The algebraic system (25) becomes singular when transitions are possible toward a degenerate state. The degeneracy problem can easily be solved numerically with the Lagrangemesh method by removing the degenerate eigenvalue. When solving system (25), the inverse
of the matrix in the left-hand member is replaced according to
(Hl′ − En N )−1 →
X
′ (k)
v
k
(E (k) − En )−1 v (k)T
(27)
where E (k) and v (k) are the eigenvalues and eigenvectors of the generalized eigenvalue problem (Hl′ − E (k) N )v (k) = 0 and T means transposition. The prime in the sum indicates that
the term with |E (k) − En | < ǫ is dropped, where ǫ is for example chosen as 10−10 in double
precision.
A standard Lagrange-mesh calculation would involve the Gauss approximation everywhere, i.e. the replacements
Nij → δij and Tij → TijG
(28)
in Eqs. (22), (23), (25), and (27). We shall see that this approximation essentially leads to
the same results in spite of the Gauss approximation. In fact, we shall even observe that
it is more accurate. The propagation of rounding errors is indeed less important in the
standard Lagrange-mesh calculation, probably because the overlap matrix N is diagonal
and the eigenvalue problem is not generalized.
IV.
NUMERICAL AND DEDUCED ANALYTICAL DIPOLE POLARIZABILI-
TIES
For dipole polarizabilities, the subscripts λ = 1 and µ = 0 are dropped. Expression (7)
is usually written as
(nl)
α(nlm) = α0
(nl)
+ α2
8
3m2 − l(l + 1)
,
l(2l − 1)
(29)
(nl)
where α0
(nl)
and α2
are known as the scalar and tensor polarizabilities, respectively [3]. The
(nl)
average polarizability defined in Eq. (9) is the scalar polarizability α0 . Exact expressions
(nl)
for α0
(nl)
and α2
will be deduced from accurate numerical results.
As we have seen, when 2h takes an integer value n, the wave function ψnl given by (21) is
exact. The exact wave functions in the right-hand member of (6) are exponentials exp(−r/n)
multiplied by polynomials. The exact solution of an inhomogeneous equation such as (6)
is also a polynomial times the same exponential (see examples in Ref. [6]). Hence, this
(1)
equation can also be solved exactly, i.e. the coefficients cnll′ j obtained from the system (25)
(1)
give the exact functions ψnll′ (r) when h = n/2. The reduced polarizabilities (8) are then
obtained with the Gauss quadrature (26) which is exact for N ≥ n + λ + 1. With an infinite
number of digits, the polarizability would be exact. Here they will contain rounding errors
related to the double-precision computer accuracy.
The variational energies for n ≤ 4 are presented in Table I. The scalar and tensor dipole
polarizabilities obtained with the variational calculation are also displayed in Table I. The
conditions of the calculation are N = n + 2 and h = n/2. The obtained values are close to
integers or simple fractions, which can easily be guessed. The 1s polarizability is known for a
long time [4]. For l = 0, the results agree with the exact values (2) of Ref. [9]. For l = n − 1,
they also agree with the exact values (3). The 2p0 and 3d0 polarizabilities calculated with
Eq. (29) and Table I reproduce those obtained from a numerical calculation with B splines
in Ref. [6]. It is remarkable that the Lagrange-mesh method with four and five mesh points
gives a better accuracy than eighty B-splines in Ref. [6]. Other cases do not seem to be
available in the literature.
With the standard double precision, exact scalar polarizabilities can be obtained up to
n = 30, at least, as exemplified by Table II. Strikingly, the Lagrange-mesh results, obtained
with the Gauss approximation, are more accurate than the variational ones, i.e. they are
closer to integer values. The tensor polarizabilities can be obtained with the same accuracy
and hence the polarizabilities of any state nlm as a function of m. Scalar polarizabilities
can be safely obtained up to n = 40. However, beyond n ≈ 30, multiprecision should be
used to avoid ambiguities in the non necessarily integer values of the tensor polarizabilities.
From small subsets of values, analytical expressions for the polarizabilities can easily be
extracted. These formulas can be tested with all other values. The two reduced dipole
9
nl
(nl)
h N En
(nl)
α0
α2
1s 0.5 3 −0.4999999999999997
4.500000000000
0
2s 1.0 4 −0.1249999999999999
120.000000000000
0
−0.1249999999999998
175.999999999999
−19.999999999999
3s 1.5 5 −0.05555555555555552 1012.500000000018
0
2p
3p
−0.05555555555555559 1295.999999999986 −161.999999999997
3d
−0.05555555555555564 1862.999999999988 −485.999999999997
4s 2.0 6 −0.03125000000000000 4991.999999999993
0
4p
−0.03124999999999998 5887.999999999932 −780.799999999996
4d
−0.03124999999999995 7680.000000000189 −1920.000000000067
4f
−0.03124999999999990 10367.99999999927 −3647.999999999685
TABLE I. Energies, scalar and tensor dipole polarizabilities obtained in a variational calculation
with N = n + 2 Lagrange-Laguerre basis functions and h = n/2 for principal quantum numbers
n ≤ 4.
polarizabilities read
α(nll+1) =
l+1
n4 [6(2n2 + 7) + (5n2 + 58)l + 24l2 + 3l3 ]
4(2l + 1)
(30)
l
n4 [7n2 + 5 − (5n2 + 19)l + 15l2 − 3l3 ].
4(2l + 1)
(31)
and
α(nll−1) =
They allow calculating the separate contributions of l′ = l + 1 and l′ = l − 1 for any nlm
state and µ = −1, 0, or 1.
The scalar dipole polarizability of level nl is given by three times the sum of expressions
(30) and (31),
(nl)
α0
1
= n4 [4n2 + 14 + 7l(l + 1)].
4
(32)
For l = 0, the formula (2) of McDowell [9] is recovered. See Table II for a comparison with
numerical results at n = 30. The tensor dipole polarizability of level nl reads
(nl)
α2
=−
n4 l
[3n2 − 9 + 11l(l + 1)].
4(2l + 3)
10
(33)
n
l
Variational
Lagrange mesh
Eq. (32)
30 0 731834999.9998 731835000.0002 731835000
1 734669999.9984 734670000.0000 734670000
2 740339999.9730 740340000.0000 740340000
3 748845000.0436 748845000.0001 748845000
4 760184999.9699 760184999.9999 760185000
25 1653210000.1764 1653210000.0003 1653210000
26 1726919998.8397 1726920000.0004 1726920000
27 1803465000.8468 1803464999.9983 1803465000
28 1882845000.0327 1882845000.0014 1882845000
29 1965059999.9875 1965059999.9998 1965060000
(nl)
TABLE II. Typical scalar dipole polarizabilities α0
for n = 30 levels. Comparison of variational
and Lagrange-mesh results with exact formula (32).
These results reproduce those of Krylovetsky, Manakov, and Marmo [5]: combining Eqs. (29),
(n,n−1)
(32), and (33) for m = 0 provides Eq. (4). For l = n − 1, the sum α0
(n,n−1)
+ α2
agrees
with Eq. (3). The 2p0 and 3d0 polarizabilities of Ref. [6] (216 and 2349, respectively) are
easily reproduced with Eq. (29).
V.
CONCLUDING REMARKS
The central results of this work are the analytical expressions (32) and (33). They allow a
simple calculation of the dipole polarizability for any state nlm of the hydrogen atom. The
computer-aided technique used to derive these analytical formulae is unusual. In a strict
mathematical sense, expressions (32) and (33) are not proven; they are only conjectured.
However, in a numerical sense, they are fully established since they agree with all results
obtained with a numerical method which would be exact in a computation with an infinite
number of digits. The roundings made necessary by the limited computer accuracy are
unambiguous for all states up to n = 30 at least.
This is a special example of use of the Lagrange-mesh method. To ensure the exactness of
the energies and wave functions, the calculations are performed with the exact overlap and
11
kinetic-energy matrix elements. As already noticed in other cases [13, 18], calculations with
the usual Gauss quadrature approximation for these matrix elements lead to the same results
with often a better accuracy. An excellent accuracy has also been obtained in calculations
of polarizabilities of the confined hydrogen atom [19] and the hydrogen molecular ion [20].
The originality here is that a rare case where the method is exact leads to general analytical
results.
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